Physics 2102 Jonathan Dowling Lecture 28 MON 23 MAR Ch 30 10 11 Inductors Inductance Nikolai Tesla EXAM 03 6PM THU 02 APR 2009 The exam will cover Ch 28 second half through Ch 32 1 3 displacement current and Maxwell s equations The exam will be based on HW08 HW11 Inductance Consider a solenoid of length that has N loops of B N windings per unit length A current i flows through the solenoid and generates a uniform magnetic field B 0 ni inside the solenoid area A each and n The solenoid magnetic flux is B NBA 2 L 0 n A 0 N A 0 N 2 A 2 The total number of turns N n B 0 n 2 A i The result we got for the special case of the solenoid is true for any inductor B Li Here L is a constant known as the inductance of the solenoid The inductance depends on the geometry of the particular inductor Inductance of the Solenoid B 0 n 2 Ai For the solenoid L 0 n 2 A i i RL Circuit Fluxing up an Inductor How does the current in the circuit change with time i t di iR E L 0 dt E t i t 1 e R i t Fast Small E R Slow Large Time constant of RL circuit L R t RL Circuit Fluxing UP L R i 0 0 E i 0 L VL 0 E E i R i 0 VL 0 i t E 1 e t R E t i t e L VL t Li t Ee t Fluxing Down an Inductor The switch is at a for a long time until the inductor is charged Then the switch is closed to b i What is the current in the circuit Loop rule around the new circuit di iR L 0 dt E Rt L E t i t e e R R i t Exponential defluxing E R t Inductors Energy Recall that capacitors store energy in an electric field Inductors store energy in a magnetic field di E iR L dt di 2 iE i R Li dt i P iV i2R 2 d Li 2 iE i R dt 2 Power delivered by battery power dissipated by R d dt energy stored in L Inductors Energy Li 2 UB 2 Magnetic Potential Energy UB Stored in an Inductor di P Li dt Magnetic Power Returned from Fluxing Down Inductor to Circuit or Put into Fluxing Up Inductor from Circuit Example The switch has been in position a for a long time It is now moved to position b without breaking the circuit What is the total energy dissipated by the resistor until the circuit reaches equilibrium 10 9V 10 H When switch has been in position a for long time current through inductor 9V 10 0 9A Energy stored in inductor 0 5 10H 0 9A 2 4 05 J When inductor defluxes through the resistor all this stored energy is dissipated as heat 4 05 J Energy Density of a Magnetic Field Consider the solenoid of length and loop area A B B2 uB 2 0 that has n windings per unit length The solenoid carries a current i that generates a uniform magnetic field B 0 ni inside the solenoid The magnetic field outside the solenoid is approximately zero 1 2 0 n 2 A i 2 The energy U B stored by the inductor is equal to Li 2 2 This energy is stored in the empty space where the magnetic field is present UB where V is the volume inside V 0 n 2 A i 2 0 n 2i 2 0 2 n 2i 2 B 2 the solenoid The density uB 2 A 2 2 0 2 0 We define as energy density uB This result even though it was derived for the special case of a uniform magnetic field holds true in general 3 Units u B J m Energy Density in E and B Fields 0E uE 2 2 2 B uB 2 0 The Energy Density of the Earth s Magnetic Field Protects us from the Solar Wind
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